Theorem breq12 4114

Description: Equality theorem for a binary relation. (Contributed by NM, 8-Feb-1996.)

Assertion

Ref	Expression
breq12	|- ( ( A = B /\ C = D ) -> ( A R C <-> B R D ) )

Proof of Theorem breq12

Step	Hyp	Ref	Expression
1		breq1 4112	. 2 |- ( A = B -> ( A R C <-> B R C ) )
2		breq2 4113	. 2 |- ( C = D -> ( B R C <-> B R D ) )
3	1, 2	sylan9bb 462	1 |- ( ( A = B /\ C = D ) -> ( A R C <-> B R D ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1398   class class class wbr 4109

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-v 2815  df-un 3215  df-sn 3695  df-pr 3696  df-op 3698  df-br 4110

This theorem is referenced by:  breq12i  4118  breq12d  4122  breqan12d  4125  posng  4822  isopolem  5995  poxp  6428  rbropapd  6473  ecopover  6867  ecopoverg  6870  ltdcnq  7712  recexpr  7953  ltresr  8154  reapval  8850  ltxr  10108  xrltnr  10112  xrltnsym  10126  xrlttr  10128  xrltso  10129  xrlttri3  10130  xposdif  10215  f1olecpbl  13526  wlk2f  16346  exmidsbthrlem  16802